Skip to main content
Log in

Testing for boundary conditions in case of fractionally integrated processes

  • Original Paper
  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

Bounded integrated time series are a recent development of the time series literature. In this paper, we work on testing the presence of unknown boundaries with particular attention to the class of fractionally integrated time series. We firstly show, via a preliminary Monte Carlo experiment, the effects of neglected boundaries conditions on the most commonly used estimators of the long memory parameter. Then, we develop a sieve bootstrap test to distinguish between unbounded and bounded fractionally integrated time series. We assess the finite sample performance of our test with a Monte Carlo experiment and apply it to the data set of the time series of the Danish Krone/Euro exchange rate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Trokic (2013) adopts the terms regulated and bounded as synonyms. We will follow Cavaliere’s terminology (Cavaliere 2002, 2005a; Cavaliere and Xu 2014) and adopt only the term bounded process.

  2. For clarity’s sake, we stress once more that we refer here to the overall long memory of the process \(X_t\), denoted by \(d=d'+1\), where \(d'\) is the long memory parameter of \(\varDelta X_t=Z_t\)

  3. We are well aware of the existence of many other estimation methods, but we believe that it is of interest to the reader the performance of the methods that are effectively the best-known and, also, implemented in the most common packages (e.g. R, Matlab,...).

  4. All the codes (throughout the paper) are written in R language (R Core Team 2015) and are available upon request by the authors.

  5. We concentrate only on the \(\hat{r}_{\mu }\) statistic proposed by Cavaliere (2002) and do not focus on the other versions (presented in the same article) accounting for a deterministic trend in the data, that is a case into which we are not interested.

  6. Poskitt (2008) also showed that the sieve bootstrap performs better than the block bootstrap (Kunsch 1989), that is also a well-known approach to boostrapping dependent data, but more suitable for short range dependence than long range dependence.

  7. Palm et al. (2008) also showed that for some data generating processes (not our case), residuals from a first order autoregression can lead to an even better performance of the sieve bootstrap in terms of asymptotic validity, compared to first difference.

  8. Results with the censoring algorithm are substantially equivalent and are available upon request by the authors.

  9. The euro is at the core of ERM 2, and the currencies of participating EU member states have central rates against the euro, but not against each other. The obligation to intervene–that is, to buy or sell currency to support the exchange rate–if a participating currency reaches a fluctuation limit depends only on the central bank of the relevant member state and the ECB. The other participating member states have no obligation to intervene. ERM 2 includes a provision on unlimited intervention credit between the ECB and the participating central banks in connection with intervention at the fluctuation limits. One of the convergence criteria for joining the euro area is to observe the normal fluctuation band within ERM 2 without severe tensions for at least two years. In the same period, the member state in question may not devalue its currency against the euro.

  10. The ERM 1, antecedent to the ERM 2, is one of the arrangements implied by the European Monetary System (EMS), in place between 1979 and May 1998.

References

  • Andrews D (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59:817–858

    Article  MathSciNet  Google Scholar 

  • Bartlett MS (1950) Periodogram analysis and continuous spectra. Biometrika 37:1–16

    Article  MathSciNet  Google Scholar 

  • Barkoulas JT, Barilla AG, Wells W (2016) Long-memory exchange rate dynamics in the euro era. Chaos Solitons Fractals 86:92–100

    Article  Google Scholar 

  • Bickel PJ, Bühlmann P (1999) A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5:413–446

    Article  MathSciNet  Google Scholar 

  • Blackman RB, Tukey JW (1958) The measurement of power spectra from the point of view of communications engineering—part I. Bell Syst Tech J 37:185–282

    Article  Google Scholar 

  • Bühlmann P (1997) Sieve bootstrap for time series. Bernoulli 3:123–148

    Article  MathSciNet  Google Scholar 

  • Cavaliere G (2002) Bounded integrated processes and unit root tests. Stat Methods Appl 11:41–69

    Article  Google Scholar 

  • Cavaliere G (2005) Limited time series with a unit root. Econ Theory 21:907–945

    MathSciNet  MATH  Google Scholar 

  • Cavaliere G (2005) Testing mean reversion in target-zone exchange rates. Appl Econ 37:2335–2347

    Article  Google Scholar 

  • Cavaliere G, Xu F (2014) Testing for unit roots in bounded time series. J Econ 178:259–272

    Article  MathSciNet  Google Scholar 

  • Chang Y, Park JY (2003) A sieve bootstrap for the test of a unit root. J Time Ser Anal 24:379–400

    Article  MathSciNet  Google Scholar 

  • Dahlhaus R (1989) Efficient parameter estimation for self-similar processes. Ann Stat 17:1749–1766

    Article  MathSciNet  Google Scholar 

  • Dickey DA, Fuller WA (1979) Distribution of the estimators of an autoregressive time series with a unit root. J Am Stat Assoc 74:427–431

    MathSciNet  MATH  Google Scholar 

  • Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26

    Article  MathSciNet  Google Scholar 

  • Fox R, Taqqu MS (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann Stat 14:517–532

    Article  MathSciNet  Google Scholar 

  • Geweke J, Porter-Hudack S (1983) The estimation and application of long-memory time series models. J Time Ser Anal 4:221–237

    Article  MathSciNet  Google Scholar 

  • Granger CWJ (2010) Some thoughts on the development of cointegration. J Econ 158:3–6

    Article  MathSciNet  Google Scholar 

  • Hurst H (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civil Eng 116:770–799

    Google Scholar 

  • Hurvich C, Ray B (1995) Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. J Time Ser Anal 16:17–41

    Article  MathSciNet  Google Scholar 

  • Kapetanios G, Psaradakis Z (2006) Sieve bootstrap for strongly dependent stationary processes. Working Papers 552, Queen Mary University of London. School of Economics and Finance

  • Kreiss JP (1992) Bootstrap procedures for AR(\(\infty \))-processes. In: Lecture Notes in Economics and Mathematical Systems, vol 376: 107–113 (Proc. Bootstrapping and Related Techniques, Trier)

  • Kunsch HR (1989) The jacknife and the bootstrap for general stationary observations. Ann Stat 17:1217–1241

    Article  Google Scholar 

  • Lahiri SN (2003) Resampling Methods for Dependent Data. Springer, New York

    Book  Google Scholar 

  • Lo A (1991) Long-term memory in stock market prices. Econometrica 59:1279–1313

    Article  Google Scholar 

  • Mandelbrot B (1972) Statistical methodology for nonperiodic cycles: from the covariance to r/s analysis. Ann Econ Soc Meas 1:259–290

    Google Scholar 

  • Mandelbrot B (1975) Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z Wahr verw Geb 31:271–285

    Article  MathSciNet  Google Scholar 

  • Newey WK, West KD (1987) A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708

    Article  MathSciNet  Google Scholar 

  • Palm FC, Smeekes S, Urbain JP (2008) Bootstrap unit-root tests: comparison and extensions. J Time Ser Anal 29:371–400

    Article  MathSciNet  Google Scholar 

  • Paparoditis E (1996) Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. J Multivar Anal 57:277–296

    Article  MathSciNet  Google Scholar 

  • Parzen E (1961) An approach to time series analysis. Ann Math Stat 32:951–989

    Article  Google Scholar 

  • Perron P, Ng S (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Rev Econ Stud 63:435–463

    Article  Google Scholar 

  • Phillips PCB, Perron P (1988) Testing for unit root in time series regression. Biometrika 75:335–346

    Article  MathSciNet  Google Scholar 

  • Poskitt DS (2008) Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. J Time Ser Anal 29:224–250

    Article  MathSciNet  Google Scholar 

  • Priestley MB (1962) Basic considerations in the estimation of spectra. Technometrics 4:551–564

    Article  MathSciNet  Google Scholar 

  • Psaradakis Z (2001) Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors. J Time Ser Anal 22:577–594

    Article  MathSciNet  Google Scholar 

  • R Core Team (2015) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria

  • Robinson PM (1995a) Log-periodogram regression of time series with long range dependence. Ann Stat 23:1048–1072

    Article  MathSciNet  Google Scholar 

  • Robinson PM (1995b) Gaussian semiparametric estimation of long range dependence. Ann Stat 23:1630–1661

    Article  MathSciNet  Google Scholar 

  • Trokic M (2013) Regulated fractionally integrated processes. J Time Ser Anal 34:591–601

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very thankful to the Editor and two anonymous Referees for the helpful and constructive comments on a previous version of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Margherita Gerolimetto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gerolimetto, M., Magrini, S. Testing for boundary conditions in case of fractionally integrated processes. Stat Methods Appl 29, 357–371 (2020). https://doi.org/10.1007/s10260-019-00474-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-019-00474-w

Keywords

JEL Classifications

Navigation